Mentor: Ðorđe Radičević
Quantum chaos is a field that studies universal properties of spectra of quantum systems. Very roughly, one can distinguish two universal types of states in a spectrum: localized (integrable) and delocalized (chaotic). Our understanding of this rough dichotomy as a function of the quantum-mechanical potential is not complete, but is rather important for a deep understanding of quantum mechanics. In particular, the behavior of systems near localization/delocalization phase transitions is poorly understood. Therefore, in this paper we investigate the spectrum of a quantum particle with conformal symmetry and compare it with that of the well- -known Aubry-André model, which is known to exhibit a delocalization phase transition. We analyse the spectra of these models in various ways, and we introduce a new diagnostic – the density of diversities – that clearly shows the difference between two systems. Our results suggest that, contrary to naive expectations, conformal quantum mechanics does not emerge at the localization/delocalization phase transition.